But it applies equally to any spherically symmetric objects, where r is the distance between the centers of mass of those objects. Note that strictly speaking, Equation 13.1 applies to point masses-all the mass is located at one point. These equal but opposite forces reflect Newton’s third law, which we discussed earlier. In later chapters, we see this same form in the electromagnetic force.įigure 13.2 Gravitational force acts along a line joining the centers of mass of two objects. The surface area of that sphere is proportional to r 2 r 2. The strength of any source at a distance r is spread over the surface of a sphere centered about the mass. We now know that this inverse square law is ubiquitous in nature, a function of geometry for point sources. He could explain this by postulating that a force exists between any two objects, whose magnitude is given by the product of the two masses divided by the square of the distance between them. Newton noted that objects at Earth’s surface (hence at a distance of R E R E from the center of Earth) have an acceleration of g, but the Moon, at a distance of about 60 R E 60 R E, has a centripetal acceleration about ( 60 ) 2 ( 60 ) 2 times smaller than g. The consequences of his theory gave rise to many remarkable predictions, essentially all of which have been confirmed over the many decades following the publication of the theory (including the 2015 measurement of gravitational waves from the merger of two black holes). His theory comes from a vastly different perspective, in which gravity is a manifestation of mass warping space and time. She also utilized calculus to explain gravity, which helped lead to its acceptance.įinally, in Einstein’s Theory of Gravity, we look at the theory of general relativity proposed by Albert Einstein in 1916. Du Châtelet, who had earlier laid the foundation for the understanding of conservation of energy as well as the principle that light had no mass, translated and augmented Newton's key work. It took the work of another prominent philosopher, writer, and scientist, Émilie du Châtelet, to establish the Newtonian gravitation as the accurate and overarching law. Other prominent scientists and mathematicians of the time, particularly those outside of England, were reluctant to accept Newton's principles. However, it was Isaac Newton who connected the acceleration of objects near Earth’s surface with the centripetal acceleration of the Moon in its orbit about Earth. Kepler showed that the motion of each planet is an ellipse (the first of his three laws, discussed in Kepler’s Laws of Planetary Motion), and Robert Hooke (the same Hooke who formulated Hooke’s law for springs) intuitively suggested that these motions are due to the planets being attracted to the Sun. This idea was supported by the incredibly precise naked-eye measurements of planetary motions by Tycho Brahe and their analysis by Johannes Kepler and Galileo Galilei. Nicolaus Copernicus (1473–1543) is generally credited as being the first to produce a mathematical model of a heliocentric system and to challenge Ptolemy’s geocentric system. The idea of heliocentrism, in which the Sun is at the center of the solar system, dates at least as far back as Aristarchus of Samos in the third century BCE, but attracted little attention, and Ptolemy’s geocentric (Earth-centered) model held sway in Europe until the sixteenth century. However, there is little evidence that anyone connected the motion of astronomical bodies with the motion of objects falling to Earth-until the seventeenth century.

These motions were described with amazing accuracy by Ptolemy (90–168 CE), whose method of epicycles described the paths of the planets as circles within circles. The motions of the Sun, our Moon, and the planets have been studied for thousands of years as well. Brahmagupta (598~665 CE) postulated that Earth was a sphere and that objects possessed a natural affinity for it, falling toward the center from wherever they were located. Aristotle (384–322 BCE) believed that it was the nature of rocks to seek Earth and the nature of fire to seek the Heavens. The earliest philosophers wondered why objects naturally tend to fall toward the ground. Then we examine the simplest form of Newton’s law of universal gravitation and how to apply it. We first review the history of the study of gravitation, with emphasis on those phenomena that for thousands of years have inspired philosophers and scientists to search for an explanation. Estimate the gravitational force between collections of mass.Calculate the gravitational force between two point masses.List the significant milestones in the history of gravitation.By the end of this section, you will be able to:

Physics phenomena gravity